If a + b + c + d = 4 then find the value of 1/(1-a)(1-b)(1-c) + 1/(1-b)(1-c)(1-d) + 1/(1-c)(1-d)(1-a) + 1/(1-d)(1-b)(1-a).

We need to reduce the given expression and then add the value of (a + b + c + d)

Answer: If a + b + c + d = 4, the value of 1/(1-a)(1-b)(1-c) + 1/(1-b)(1-c)(1-d) + 1/(1-c)(1-d)(1-a) + 1/(1-d)(1-b)(1-a) is 0

Let's simplify 1/(1-a)(1-b)(1-c) + 1/(1-b)(1-c)(1-d) + 1/(1-c)(1-d)(1-a) + 1/(1-d)(1-b)(1-a)

Explanation:

Here, we need to take LCM of the denominators which is equal to (1-a)(1-b)(1-c)(1-d)

Then we need to simplify further:

= [(1-d) + (1-a) + (1-b) + (1-c)] / [(1-a)(1-b)(1-c)(1-d)]

= [4 - (a + b + c + d)] / [(1-a)(1-b)(1-c)(1-d)]

= [4 - 4] / [(1-a)(1-b)(1-c)(1-d)]  {since a + b + c + d = 4}

= 0

Thus, the value of 1/(1-a)(1-b)(1-c) + 1/(1-b)(1-c)(1-d) + 1/(1-c)(1-d)(1-a) + 1/(1-d)(1-b)(1-a) is 0, if a + b + c + d = 4